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11.6 Solving Systems with Gaussian Elimination

Augmented matrices, the three row operations, and reduction to row-echelon form so the solution reads straight off.

Algebra and Trigonometry 2e, 11.6 Solving Systems with Gaussian Elimination

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48 exercises

Writing the Augmented Matrix for a System of Equations

Write the augmented matrix for the given system of equations.

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11.6 Exercise 8

Write the augmented matrix of the given system of equations.

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Writing a System of Equations from an Augmented Matrix Form

Find the system of equations from the augmented matrix.

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11.6 Exercise 10

Write the system of equations from the augmented matrix.

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Solving a 2 × 2 System by Gaussian Elimination

Solve the given system by Gaussian elimination.

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11.6 Exercise 12

Solve the given system by Gaussian elimination.

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Using Gaussian Elimination to Solve a System of Equations

Use Gaussian elimination to solve the given 2×2 2 \times 2 system of equations.

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Solving a Dependent System

Solve the system of equations.

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Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form

Perform row operations on the given matrix to obtain row-echelon form.

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11.6 Exercise 16

Write the system of equations in row-echelon form.

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Solving a System of Linear Equations Using Matrices

Solve the system of linear equations using matrices.

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Solving a Dependent System of Linear Equations Using Matrices

Solve the following system of linear equations using matrices.

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11.6 Exercise 19

Solve the system using matrices.

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Solving Systems of Equations with Matrices Using a Calculator

Solve the system of equations.

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Applying 2 × 2 Matrices to Finance

Carolyn invests a total of $12,000 in two municipal bonds, one paying 10.5% interest and the other paying 12% interest. The annual interest earned on the two investments last year was $1,335. How much was invested at each rate?

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Applying 3 × 3 Matrices to Finance

Ava invests a total of $10,000 in three accounts, one paying 5% interest, another paying 8% interest, and the third paying 9% interest. The annual interest earned on the three investments last year was $770. The amount invested at 9% was twice the amount invested at 5%. How much was invested at each rate?

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11.6 Exercise 23

A small shoe company took out a loan of $1,500,000 to expand their inventory. Part of the money was borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount borrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three loans was $130,500. Use matrices to find the amount borrowed at each rate.

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11.6 Exercise 24

Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

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11.6 Exercise 26

Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix [ 9 3 1 −2  |   0 6 ]. \begin{bmatrix} 9 & 3 \\ 1 & - 2 \textrm{ } \left|\right. \textrm{ } 0 \\ 6 \end{bmatrix} .

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11.6 Exercise 28

Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.

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11.6 Exercise 30

Write the augmented matrix for the linear system.

  16y=4 9x−y=2 \begin{matrix}\textrm{ }\textrm{ } 16 y = 4 \\ 9 x - y = 2\end{matrix}

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11.6 Exercise 32

Write the augmented matrix for the linear system.

 x+5y+8z=19 12x+3y=4 3x+4y+9z=−7 \begin{matrix} \\ \textrm{ } x + 5 y + 8 z = 19 \\ 12 x + 3 y = 4 \\ 3 x + 4 y + 9 z = −7\end{matrix}

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11.6 Exercise 34

Write the linear system from the augmented matrix.

[ −2 5 6 −18  |   5 26 ] \begin{bmatrix} −2 & 5 \\ 6 & −18 \textrm{ } \left|\right. \textrm{ } 5 \\ 26 \end{bmatrix}

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11.6 Exercise 36

Write the linear system from the augmented matrix.

[ 3 2 0 −1 −9 4 8 5 7  |   3 −1 8 ] \begin{bmatrix} 3 & 2 & 0 \\ −1 & −9 & 4 \\ 8 & 5 & 7 \textrm{ } \left|\right. \textrm{ } 3 \\ −1 \\ 8 \end{bmatrix}

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11.6 Exercise 38

Write the linear system from the augmented matrix.

[ 4 5 −2 0 1 58 8 7 −3  |   12 2 −5 ] \begin{bmatrix} 4 & 5 & −2 \\ 0 & 1 & 58 \\ 8 & 7 & −3 \textrm{ } \left|\right. \textrm{ } 12 \\ 2 \\ −5 \end{bmatrix}

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11.6 Exercise 40

Solve the system by Gaussian elimination.

[ 1 0 1 0  |   1 2 ] \begin{bmatrix} 1 & 0 \\ 1 & 0 \textrm{ } \left|\right. \textrm{ } 1 \\ 2 \end{bmatrix}

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11.6 Exercise 42

Solve the system by Gaussian elimination.

[ −1 2 4 −5  |   −3 6 ] \begin{bmatrix} −1 & 2 \\ 4 & −5 \textrm{ } \left|\right. \textrm{ } −3 \\ 6 \end{bmatrix}

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11.6 Exercise 44

Solve the system by Gaussian elimination.

 2x−3y=−9 5x+4y=58 \begin{matrix}\textrm{ } 2 x - 3 y = - 9 \\ 5 x + 4 y = 58\end{matrix}

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11.6 Exercise 46

Solve the system by Gaussian elimination.

2x+3y=12  4x+y=14 \begin{matrix}2 x + 3 y = 12 \\ \textrm{ } 4 x + y = 14\end{matrix}

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11.6 Exercise 48

Solve the system by Gaussian elimination.

−5x+8y=3 10x+6y=5 \begin{matrix}−5 x + 8 y = 3 \\ 10 x + 6 y = 5\end{matrix}

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11.6 Exercise 50

Solve the system by Gaussian elimination.

−60x+45y=12  20x−15y=−4 \begin{matrix}−60 x + 45 y = 12 \\ \textrm{ } 20 x −15 y = −4\end{matrix}

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11.6 Exercise 52

Solve the system by Gaussian elimination.

2x−y=2 3x+2y=17 \begin{matrix} 2 x - y = 2 \\ 3 x + 2 y = 17\end{matrix}

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11.6 Exercise 54

Solve the system by Gaussian elimination.

3 4 x− 3 5 y=4 1 4 x+ 2 3 y=1 \begin{matrix}\frac{3}{4} x - \frac{3}{5} y = 4 \\ \frac{1}{4} x + \frac{2}{3} y = 1\end{matrix}

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11.6 Exercise 56

Solve the system by Gaussian elimination.

[ 1 0 0 0 1 1 0 0 1  |   31 45 87 ] \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \textrm{ } \left|\right. \textrm{ } 31 \\ 45 \\ 87 \end{bmatrix}

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11.6 Exercise 58

Solve the system by Gaussian elimination.

[ 1 2 3 0 5 6 0 0 8  |   4 7 9 ] \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 8 \textrm{ } \left|\right. \textrm{ } 4 \\ 7 \\ 9 \end{bmatrix}

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11.6 Exercise 60

Solve the system by Gaussian elimination.

−2x+3y−2z=3      4x+2y−z=9 4x−8y+2z=−6 \begin{matrix} −2 x + 3 y - 2 z = 3 \\ \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 4 x + 2 y - z = 9 \\ 4 x - 8 y + 2 z = −6\end{matrix}

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11.6 Exercise 62

Solve the system by Gaussian elimination.

     2x+3y+2z=1  −4x−6y−4z=−2 10x+15y+10z=5 \begin{matrix}\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } 2 x + 3 y + 2 z = 1 \\ \textrm{ } −4 x - 6 y - 4 z = −2 \\ 10 x + 15 y + 10 z = 5\end{matrix}

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11.6 Exercise 64

Solve the system by Gaussian elimination.

   x+2y−z=1 −x−2y+2z=−2 3x+6y−3z=3 \begin{matrix}\textrm{ }\textrm{ }\textrm{ } x + 2 y - z = 1 \\ - x −2 y + 2 z = −2 \\ 3 x + 6 y −3 z = 3\end{matrix}

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11.6 Exercise 66

Solve the system by Gaussian elimination.

x+y+z=100    x+2z=125 −y+2z=25 \begin{matrix}x + y + z = 100 \\ \textrm{ }\textrm{ }\textrm{ } x + 2 z = 125 \\ - y + 2 z = 25\end{matrix}

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11.6 Exercise 68

Solve the system by Gaussian elimination.

− 1 2 x+ 1 2 y+ 1 7 z=− 53 14    1 2 x− 1 2 y+ 1 4 z=3     1 4 x+ 1 5 y+ 1 3 z= 23 15 \begin{matrix}- \frac{1}{2} x + \frac{1}{2} y + \frac{1}{7} z = - \frac{53}{14} \\ \textrm{ }\textrm{ } \frac{1}{2} x - \frac{1}{2} y + \frac{1}{4} z = 3 \\ \textrm{ }\textrm{ }\textrm{ } \frac{1}{4} x + \frac{1}{5} y + \frac{1}{3} z = \frac{23}{15}\end{matrix}

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11.6 Exercise 70

Use Gaussian elimination to solve the system.

x−1 7 + y−2 8 + z−3 4 =0 x+y+z=6 x+2 3 +2y+ z−3 3 =5 \begin{matrix}\frac{x −1}{7} + \frac{y −2}{8} + \frac{z −3}{4} = 0 \\ x + y + z = 6 \\ \frac{x + 2}{3} + 2 y + \frac{z −3}{3} = 5\end{matrix}

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11.6 Exercise 72

Use Gaussian elimination to solve the system.

x−3 4 − y−1 3 +2z=−1 x+5 2 + y+5 2 + z+5 2 =8 x+y+z=1 \begin{matrix}\frac{x −3}{4} - \frac{y −1}{3} + 2 z = −1 \\ \frac{x + 5}{2} + \frac{y + 5}{2} + \frac{z + 5}{2} = 8 \\ x + y + z = 1\end{matrix}

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11.6 Exercise 74

Use Gaussian elimination to solve the system.

x−3 4 − y−1 3 +2z=−1 x+5 2 + y+5 2 + z+5 2 =7 x+y+z=1 \begin{matrix}\frac{x −3}{4} - \frac{y −1}{3} + 2 z = −1 \\ \frac{x + 5}{2} + \frac{y + 5}{2} + \frac{z + 5}{2} = 7 \\ x + y + z = 1\end{matrix}

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11.6 Exercise 76

Set up the augmented matrix that describes the situation, and solve for the desired solution.

At Bakari's competing cupcake store, $4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost $2.25 and the red velvet cupcakes cost $1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day?

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11.6 Exercise 78

Set up the augmented matrix that describes the situation, and solve for the desired solution.

You invested $2,300 into account 1, and $2,700 into account 2. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

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11.6 Exercise 80

Set up the augmented matrix that describes the situation, and solve for the desired solution.

A major appliance store has agreed to order vacuums from a startup founded by college engineering students. The store would be able to purchase the vacuums for $86 each, with a delivery fee of $9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are s…

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11.6 Exercise 82

Set up the augmented matrix that describes the situation, and solve for the desired solution.

At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin s…

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11.6 Exercise 84

Set up the augmented matrix that describes the situation, and solve for the desired solution.

A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of ea…

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.